The Multiscale Nature of Streamers
Transcript of The Multiscale Nature of Streamers
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1. Introduction
Streamer discharges are a fundamental physical phenomenon with many technical
applications [1, 2, 3]. While their role in lightning waits for further studies [4, 5],
new lightning related phenomena above thunderclouds have been discovered in the past15 years to which streamer concepts can be directly applied [6, 7, 8, 9].
In past years, appropriate methods have become available to study and
analyze these phenomena. Methods include plasma diagnostic methods, large scale
computations as well as analysis of nonlinear fronts and moving boundaries. The aim
of the present article is to briefly summarize progress in these different disciplines, to
explain the mutual benefit and to give a glimpse on future research questions.
The overall challenge in the field is to understand the growth of single streamers
as well as the conditions of branching or extinction and their interactions which would
allow us to predict the overall multi-channel structure formed by a given power supplyin a given gas. We remark that so-called dielectric breakdown models [10, 11] have been
suggested to address this question, but they incorporate the underlying mechanisms on
smaller scales in a too qualitative way.
Many of our arguments are qualitative as well, but in a different sense. The
main theoretical interest of the present paper is in basic conservation laws, in the wide
range of length and time scales that characterize a streamer, in physical mechanisms
for instabilities, and in the question whether a given problem should be modelled in
a continuous or a discrete manner. Answering these questions provides the basis for
future quantitative predictions.
The paper is organized as follows: In Section 2 a short overview over currentexperimental questions, methods and results is given, and applications are briefly
discussed. Sprite discharges above thunderclouds are reviewed. In Section 3, the
microscopic mechanisms and modelling issues are discussed and characteristic scales
are identified by dimensional analysis. In Section 4 numerical solutions for negative
streamers in non-attaching gases are presented, the multiple scales of the process are
discussed and a short view on numerical adaptive grid refinement is given. Section
5 summarizes an analytical model reduction to a moving boundary problem, sketches
issues of charge conservation and transport and confronts two different concepts of
streamer branching. Conclusion and summary can be found in Section 6.
2. Streamers and sprites, experiments, applications and observations
The emergence and propagation of streamers has a long research history. The basics of
a theory of spark breakdown were developed in the 1930ies by Raether, Loeb and Meek
[12, 13].
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2.1. Time resolved streamer measurements
The first experiments on streamers were carried out by Raether who took pictures of the
development of a streamer in a cloud chamber [12]. In this experiment a discharge was
generated by a short voltage pulse. The ions within the streamer region act as nucleifor water droplets that form in the cloud chamber. Photographs of these droplets show
the shape of the avalanche or the emerging streamer.
Later the use of streak photography together with image intensifiers enabled
researchers to take time resolved pictures of streamers [14, 15, 16]. Streak photographs
show the evolution of a slit-formed section of the total picture as a function of time.
Examples of streak photographies can be found in this volume [5].
Recently, ICCD (Intensified Charge Coupled Device) pictures yield very high
temporal resolution, and at the same time a full picture of the discharge, rather than
the one-dimensional subsection obtained with streak photography. First measurements
since 1994 with 30 ns [17] and 5 ns [18] resolution showed the principle. Since 2001, a
resolution of about 1 ns has been reached [19, 20, 21, 22, 23, 24]. Meanwhile, even shorter
gate times are possible [25]. However, the C-B transition of the second positive system
of N2 is the most intensive and dominates the picture, and its lifetime is of the order
of 1 to 2 ns at atmospheric pressure. Therefore, a further improvement of the temporal
resolution of the camera does not improve the temporal resolution of the picture, but
has to be payed with a lower spatial resolution and a lower photon number density.
Our measurements with resolution down to about 1 ns therefore resolve the short time
structure of streamers at atmospheric pressure down to the physical limit. Fig. 1 shows
snapshots of positive streamers in ambient air emerging from a positive point electrodeat the upper left corner of the picture and extending to a plane electrode at the lower
end of the picture. The distance between point and plane is 4 cm and the applied voltage
about 28 kV. The optics resolves all streamers within the 3D discharge, also within the
depth. The filamentary structure of the streamers as well as their frequent branching is
clearly seen on the left most picture with 300 ns exposure time. The rightmost picture
has the shortest exposure time of 1 ns. It shows not the complete streamers, but only the
actively growing heads of the channels where field and impact ionization rates are high.
As a consequence, the other pictures have to be interpreted not as glowing channels,
but as the trace of the streamer head within the exposure time. Streamer velocities can
therefore directly be determined as trace length devided by exposure time.Most experimental work has been carried out on positive streamers in air [20, 24,
25, 26]. We wish to draw the readers attention to the work by Yi and Williams [ 23], who
investigated the propagation of both anode and cathode-directed streamers, in almost
pure N2 and in N2/O2 mixtures. We will briefly interprete their results in Section 3.
The experimental results depend not only on applied voltage, but also on further
features of the external power supply. A glimpse is given in Fig. 2. For a further
discussion of this feature, we refer to [24, 25, 26] and future analysis. Furthermore the
results depend on the gas pressure [24] as will be further discussed below.
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Figure 1. ICCD photographs in the present Eindhoven experiments of positive
streamers in ambient air between point electrode in the upper left corner and planeelectrode below. Only the right halves of the figures are shown. The gap spacing is
4 cm, the applied voltage is 28 kV. The exposure times of the photographs is 300,
50, 10 and 1 ns, the actual time intervals are given in brackets where t = 0 is the
approximate time when the streamers emerge from the upper point.
2.2. Applications
There are numerous applications of streamers in corona discharges [1]. Dust
precipitators use DC corona to charge small particles and draw them out of a gas
stream. This process is used in industry for already more than a century. Another widespread application is charging photoconductors in copiers and laser printers. The first
use of a pulsed discharge has been the production of ozone with a barrier configuration
in 1854. This method is still being used [1] but pulsed corona discharges obtain the
same ozone yield [28].
In the 1980s the chemical activity of pulsed corona was recognized and investigated
for the combined removal of SO2, NOx and fly ash [29, 30]. In the same period also the
first experiments on water cleaning by pulsed corona were performed [31]. More results
on combined SO2/NOx removal can be found in [1], recent results on degradation of
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Figure 2. ICCD photograph of a positive streamer in ambient air in the same electrode
configuration as in Fig. 1, but now powered with a transmission line transformer and
a peak voltage of 60 kV [27]. The voltage pulse lasts about 100 ns and limits theduration of the discharge; the exposure time of 20 s accounts for the jitter of the
voltage switching.
phenol in water are given in [28] and [32]. More recently, the chemical reactivity is
further explored for example in odor removal [33], tar removal from biogas [34] and
killing of bacteria in water [35].
A new field is the combination of chemical and hydrodynamic effects. This
can be used in plasma-assisted combustion [36] and flame control [37]. Purely
electrohydrodynamic forces are studied in applications such as aerodynamic flowacceleration [3, 38] for aviation and plasma-assisted mixing [39].
Basically, these applications are based on at least one of three principles: 1) the
deposition of streamer charge in the medium, 2) molecular excitations in the streamer
head that initiate chemical processes, and 3) the coupling of moving space charge regions
to gas convection [3]. The chemical applications are based on the exotic properties of
the plasma in the streamer head that acts as a self-organized reactor: a space charge
wave carries a confined amount of high energetic electrons that effectively ionize and
excite the gas molecules. It is this active region that is seen in ICCD pictures like Fig. 1.
2.3. Sprite discharges above thunderclouds
Streamers can also be observed in nature. They play a role in creating the paths of
sparks and lightning [4, 5, 40], and sensitive cameras showed the existence of so-called
sprites [6, 41, 42] and blue jets [8, 43, 44] in the higher regions of the atmosphere above
thunderclouds. With luck and experience, sprites can also be seen with naked eye.
A scheme of sprites, jets and elves as the most frequently observed lightning related
transious luminous events above lightning clouds is given in Fig. 3a.
Sprites have been observed between about 40 and 90 km height in the atmosphere.
Above about 90 km, solar radiation maintains a plasma region, the so-called ionosphere.
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Figure 3. a) Scheme of transient luminous events above thunderclouds [45]. b)
Photograph of a sprite. The altitude is indicated on the right. Courtesy of H.C.Stenbaek-Nielsen, Dept. Geophys., Univ. Fairbanks, Alaska.
In this region, so-called elves can occur which are expanding rings created by
electromagnetic resonances in the ionospheric plasma. Sprites, on the other hand,
require a lowly or non-ionized medium, they can propagate from the ionosphere
downwards or from some lower base upwards, or they can emerge at some immediate
height and propagate upwards and downwards like in the event shown in Fig. 3b.
A variety of sprite forms have been reported [7, 46]. The propagation direction and
approximate velocity their speed can exceed 107 m/s [47] , is known since researcherssucceeded in taking movies [9]. A sequence of movie pictures can be found in this
volume [5]. Telescopic images of sprites show that they are composed of a multitude
of streamers (see Fig. 4). Blue jets propagate upwards from the top of thunderclouds,
at speeds that are typically two orders of magnitude lower than those of sprites, and
they have a characteristic conical shape and appear in a blueish color [43, 44]. Sprite
discharges are the most frequent of these phenomena.
The approximate similarity relations between streamers and sprites are discussed
in the next section. Sprites could therefore have similar physical and chemical effects
as those discussed for streamer applications in the previous section. In fact, charge
deposition in the medium and molecular excitations with subsequent chemical processes
can be expected as well. On the other hand, we will show below by dimensional analysis
that generation of gas convection is unlikely in sprites.
3. Microscopic modelling
3.1. An overview of modelling aspects
The basic microscopic ingredients for a streamer discharge are
1) the generation of electrons and ions in regions of high electric field,
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Figure 4. Telescopic image of a sprite discharge. The right photograph zooms into
the white rectangle of the left photograph. These pictures are taken from [7].
2) drift and diffusion of the electrons in the local field, and
3) the modification of the externally applied field by the generated space charges.
While avalanches evolve in a given background field, streamers have a characteristic
nonlinear coupling between densities and fields: The space charges change the field, and
the field determines the drift and reaction rates. More specifically, the streamer creates
a self-consistent field enhancement at its tip which allows it to penetrate into regions
where the background field is too low for an efficient ionization reaction to take place.
In this sense, streamers are similar to mechanical fractures in solid media: either the
electric or the mechanical forces focus at the tips of the extending structures.
While these general features are the same, models vary in the following aspects: the number of species and reactions included in a model [48, 49, 50, 53],
the choice for fluid models with continuous particle densities [48, 50, 51, 52, 53, 54]
versus models that trace single particles or super-particles [55, 56]
local versus nonlocal modelling of drift and ionization rates (by electron impact and
photo-ionization), and
assumptions about background ionization from natural radioactivity or from previous
streamer events in a pulsed streamer mode, and choices of electrode configuration,
plasma-electrode interaction, initial ionization seed and external circuit [48, 49, 50, 57,
58].
There are so-called 1.5-dimensional models that include assumptions about radialproperties into an effectively 1D numerical calculations [59, 60], so-called 2D models,
that solve the 3D problem assuming cylindrical symmetry [48, 49, 50, 51, 52, 53, 54],
and a few results on fully 3D models have been reported [61, 58].
It also should be recalled, that the result of a numerical computation does not
necessarily resemble the solution of the original equations: results can depend on spatial
grid spacing and time stepping, on the computational scheme etc. Adding many more
species with not well-known reaction rates might force a computation to use a lower
spatial resolution and lead to worse results than a reduced model. Model reduction
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techniques therefore should be applied to problems with a complex spatio-temporal
structure like streamers; they can be based on a large difference of inherent length and
time scales, key techniques are adiabatic elimination, singular perturbation theory etc.
Furthermore, considering a single streamer with cylindrical symmetry, the results of a
2D calculation will be numerically much more accurate than those of a 3D calculation,
and a fluid or continuum approximation should be sufficient. On the other hand, a
really quantitative model of streamer branching should include single-particle statistics
(not super-particles!) in the leading edge of the 3D ionization front. All in all, it can be
concluded that the model choice also depends on the physical questions to be addressed.
3.2. The minimal model
All quantitative numerical results in the present paper are obtained for negative
streamers with local impact ionization reaction in a fluid approximation for three species
densities: the electron density ne and the densities of positive and negative ions ncoupled to an electric field E in electrostatic aproximation E = R. The model
reads
ne = De2
Rne + R (e ne E)
+ (e |E| (|E|) a) ne,
n+ = e E(E) ne,
n = a ne,
2R
=e
0(ne + n n+) , E = R. (1)
Here De and a are the electron diffusion coefficient and the electron attachment rate,
e and De are the electron mobility and diffusion constant. We assume that the
impact ionization rate (E) is a function of the electric field, and that it defines
characteristic scales for cross-section 0 and field strength E0, similarly as in the
Townsend approximation (|E|) = 0 exp(E0/|E|). (The Townsend approximation
together with a = 0 is actually used in our presented computational results.) Electrons
drift in the field and diffuse, while positive and negative ions are considered to be
immobile on the time scales investigated in this paper.
3.3. Dimensional analysis
Dimensionless parameters and fields are introduced as
r = 0 R, t = 0eE0 ,
=e ne
00E0, =
e (n+ n)
00E0,
E =E
E0, f(|E|, ) = |E|
(|E| E0)
0 ,
D =De0eE0
, =a
0eE0, (2)
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which brings the system of equations (1) into the dimensionless form
t = D2 + (E) + f(|E|, ) , (3)
t = f(|E|, ) , (4)
2 = E = . (5)
It is interesting to note that even for several charged species ne and n, the computations
can be reduced to only two density fields: one for the mobile electrons and one for
the space charge density of all immobile ions .
The intrinsic parameters depend on neutral particle density N and temperature T,
for N2 they are with the parameter values from [48, 49] given by
10 2.3 m
N/N0, E0 200
kV
cm
N
N0, (6)
eE0 760
km
s = 760
m
ns , D 0.12
T
T0 , (7)00E0
e
5 1014
cm3
N
N0
2= 2 105
N
N0N, (8)
where N0 and T0 are gas density and temperature under normal conditions.
Based on this dimensional analysis, it can be stated immediately that the ionization
within the streamer head will be of the order of 1014/cm3(N/N0)2, the velocity will be
of the order 1000 km/s etc. The natural units for length, time, field and particle density
depend on gas density N, the dimensionless diffusion coefficient depends on temperature
T through the Einstein relation, and the characteristic streamer velocity is independent
of both N and T. As far as the minimal model is applicable, these equations identify thescaling relation between streamers and sprites. While lab streamers propagate at about
atmospheric pressure, sprites at a height of 70 km propagate through air with density
about 5 orders of magnitude lower. Sprite streamers are therefore about 5 decades larger
(one streamer-cm corresponds to one sprite-km etc.), but their velocities are similar.
It should be noted that all particle interactions taken into account within the
minimal model are two particle collisions between one electron or ion and one neutral
particle, therefore the intrinsic parameters simply scale with neutral particle density
N. Applying the minimal model to a different gas type at different temperature or
density can simply be taken into account by adapting the intrinsic scales (6)(8), while
the dimensionless model (3)(5) stays unchanged. This is true for the fast two particleprocesses in the front, for corrections compare, e.g., [62, 63].
3.4. Additional mechanisms: photo and background ionization, gas convection
The present model is suitable to describe negative or anode directed streamers in
gases with negligible photo-ionization like pure nitrogen. Negative streamers move
in the same direction as the electrons drift. It should be noted that they hardly
have been studied experimentally in recent years except by Yi and Williams [23]
who carefully studied the influence of small oxygen concentrations on streamers in
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nitrogen. Their negative streamers approach an oxygen-independent limit when the
oxygen concentration becomes sufficiently small. This supports our view that the
minimal model suffices to describe negative streamers in pure nitrogen.
However, positive streamers in nitrogen do depend even on very small oxygen
concentrations [23]. This can also be understood: The head of a positive or cathode
directed streamer would be depleted from electrons within the minimal model and hardly
move while a negative streamer keeps propagating through electron drift [64, 65]. On the
other hand, photo-ionization or a substantial amount of background ionization supply
electrons ahead of the streamer tip and do allow a positive streamer to move. A recent
discussion of the relative importance of these two mechanisms has been given in [58]. The
main conclusion is that in repetitive discharges, the background ionization is important.
A very interesting feature contained in none of these models is gas convection.
Typically, it is assumed that the relative density of charged particles ne/N is so small
according to dimensional analysis (8), it is of the order of 105
under normal conditions that the neutral gas stays at rest. However, streamer induced gas convection recently
has become a relevant issue in airplane hydrodynamics! Actually, streamers might
efficiently accelerate the air in the boundary layer above a wing and therefore decrease
the velocity gradient and allow the flow over the wing to stay laminar. For these exciting
results with relevance also for general streamer studies, we refer to [3, 38]. We finally
remark that this effect should not be relevant for sprite discharges, as the characteristic
degree of ionization is 2 105 N/N0 according to dimensional analysis, so it decreases
to the order of 1010 at 70 km height.
4. Numerical solutions of the minimal model
4.1. The stages of streamer evolution
The solutions of the minimal model in Fig. 5 show the characteristic states of streamer
evolution. Shown is a negative streamer in a high background field of 0.5 in dimensionless
units as presented in [51, 52, 66, 67, 68].
In the first column, a streamer just emerges from an avalanche. At this stage the
space charges are smeared out over the complete streamer head and resemble very much
the historical sketches of Raether [12] (cf. Fig. 10(a) below), as they also can be found
in the textbooks of Loeb and Meek [13] or Raizer [69].In the second column, the space charge region has contracted to a thin layer
around the head, as also can be seen in many other studies of positive or negative
streamers in nitrogen or air; now the electric field is suppressed in the ionized interior
and substantially enhanced ahead of the streamer tip. As Raethers estimate in [12]
shows (cf. Fig. 10(a)), the field cannot be substantially enhanced in the initial stage of
streamer evolution. However, it can in the second stage when the thin layer is formed.
In this stage, the enhanced field in our calculation also easily can exceed the theoretical
value suggested by Dyakonov and Kachorovskii [70, 71], and by Raizer and Simakov
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Figure 5. Evolution of the electron density distribution (upper row), net charge
density and equipotential lines (middle row) and electric field strength (lower row),
when an ionization seed attached to the cathode (z=0) is released in a background
field of 0.5 in dimensionles units. The thick lines in the lower panel indicate where
the field is higher than the background value, the thin lines where it is lower. The
snapshots are taken at dimensionless times t=50, 250 and 325. Note that the quality
and size of this figure (and some others) has been substantially reduced for the arxiv
submission and will be better in the journal publication.
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Figure 6. Sketch of the inherent scales of a propagating streamer.
[72].
In the third column, the streamer head becomes unstable and branches. Whenwe first published these results in [51], doubts were raised about the physical nature of
the branching event [73, 74], refering also to an earlier debate of a similar observation
[53, 75, 76]. This debate motivates our analysis and discussion in Section 5.
4.2. The multiscale nature of streamers
It is important to note the very different inherent scales of a propagating streamer,
even within the minimal model. They are shown in Fig. 6: there is a wide non-ionized
outer space where only the electrostatic Laplace equation 2 = 0 has to be solved to
determine the electric field. There are one or several streamer channels that are longand narrow. Around the streamer head, there is a layered structure with an ionization
region and a screening space charge region.
Furthermore, in future work more regions should be distinguished: there is the
leading edge of the front where the particle density is so low that the stochastic particle
distribution leads to substantial fluctuations, and there is the interior ionized region
where statistical fluctuations of particle densities are negligible: the characteristic
number of charged particles within a characeristic volume 30 is according to
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dimensional analysis
00E0e
30 6000
N/N0. (9)
An immediate consequence is that stochastic density fluctuations are more important forhigh pressure discharges like streamers than for sprites. Whether this leads to different
branching rates has to be investigated.
A large separation of length scales can be a benefit for analysis as it allows to use
their ratios as small parameters and to develop a ladder of reduced models, see Section
5. On the other hand, it is a major challenge for numerical calculatioms.
4.3. Adaptive grid refinement
These numerical challenges can be met by adaptive grid refinement. Such a code has
recently been constructed. It computes the evolution of a streamer on a relatively coarsegrid and refines the mesh where the fine spatial structure of the solution requires. For
preliminary and more extended results, we refer to [54, 66, 67, 68] and future papers. The
distribution of the grid at different time steps is illustrated in Fig. 7. Here the evolution
of a streamer in a long, undervolted gap is shown. More specifically, we consider a plane
parallel electrode geometry, with an inter electrode distance of approximately 65000 in
dimensionless units. The applied background electric field is uniform, and has a strength
of 0.15. For N2 under normal conditions this corresponds to a gap of about 15 cm with
a background field of 30 kV/cm.
We remark that the grid size used in the computation of the results shown in Fig. 7
is the same as that used by Vitello et al. [49] for a much smaller gap (0.5 cm) at a higherbackground field (50 kV/cm). In these simulations streamer branching was not seen,
probably due to the shortness of the gap. On the contrary, a large gap as in the above
examples enables the streamer to reach the instability even at a relatively low field.
Moreover, our previous calculations that showed streamer branching in a high
background field of 0.5, were performed on a uniform numerical grid with grid spacing
x = 2 in [51] and with x = 1 in [52]. We now are able to perform computations on
a grid that adapts locally down to x = 1/8 [67, 68].
For both background fields 0.5 and 0.15, the numerical results show that the time
of streamer branching reaches a fixed value when using finer numerical meshes. We
therefore conclude that streamer branching indeed is physical, and we will further
support this statement with different arguments in the next section. We notice that in
our cylindrically symmetric system, the streamer branches into rings, which obviously is
rather unphysical. Therefore it is not meaningful to follow the further evolution of the
streamer after branching. However, the effectively two-dimensional setting suppresses
destablizing modes that break the cylindrical symmetry, and the time of branching in a
cylindrically symmetric system therefore gives an upper bound for the branching time
in the real three-dimensional system [74].
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2000 0 20000
1000
2000
3000
t=15000
r
z
2000 0 20000
1000
2000
3000
4000
5000
t=25000
r
2000 0 20000
1000
2000
3000
4000
5000
6000
7000
8000
9000
t=33000
r
Figure 7. Evolution of the logarithm of the electron density together with the
computational grids for a streamer evolving in a background field of 0.15, with an
inter-electrode distance of approx. 65000. In pure molecular nitrogen under normal
conditions this corresponds to a 15 cm gap with a background field of 30 kV/cm.
The initial seed is attached to the cathode, allowing for a net electron inflow. The
coarsest grid, in black, has a mesh size of 64, and is refined up to a mesh size of 2
(white domains).
5. Analytical results on propagating and branching streamers
5.1. Nonlinear analysis of ionization fronts
The question of streamer branching can be addressed analytically, using concepts
developed in other branches of science: Combustion, e.g., for decades is a very active
area of applied nonlinear analysis, pattern formation and large scale computations.
Chemical species are processed/burned when fuel is available and the temperature
exceeds a threshold. The temperature is enhanced by the combustion front itself. Quite
similarly, ionization is created is there are free electrons and if the electric field exceeds
a threshold. The field is enhanced by the (curved) ionization front itself [64, 65, 51].
It is therefore attractive to develop analysis of streamers along the same lines, hencecomplementing numerical results with an analytical counterpart. This is particularly
important for addressing questions of branching, long time evolution and multi-streamer
structures. In particular, the structures of many interacting streamers in the near future
will remain numerically inaccessible without model reduction.
5.2. The moving ionization boundary
In recent work, we have elaborated streamer evolution on two levels of refinement: the
properties of a planar ionization front within the minimal model [64, 65, 51, 77], and
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Figure 8. The moving boundary approximation for an ideally conducting streamer.
the evolution of curved ionization boundaries [54, 78, 79]. A front solution is a solution
of the full fluid model (3)(5) zooming into the inner structure of the front. Ionization
boundaries are formulated on the outer scale where the ionization front is reduced to a
moving boundary between ionized and nonionized region.
If there is no initial ionization in the system, planar negative ionization fronts within
the minimal model move with asymptotic velocityv(E+) = |E+| + 2
D f(|E+|, ) (10)
into a field E+ immediately ahead of the front. The degree of ionization = behind
the front is a function of the field E+. For large fields, the front velocity is dominated by
the electron drift velocity v(E+) |E+|. For details about analyzing streamer fronts,
we refer to [64, 65, 51, 77].
On the outer level of ionized and non-ionized region, a simple evolution model
for the phase boundary can be formulated as shown in Fig. 8: assume the Lozansky
Firsov approximation [80] that the streamer interior (indicated with an upper index )
is equipotential:
= const. The exterior is free of space charges, hence 2
+
= 0.Every piece of the boundary moves with the local velocity v(E+) determined by the
local field E+ ahead of the front.
If one assumes that the electric potential across the ionization boundary is
continuous + = everywhere along the boundary, one arrives at a model that has
been studied previously in the hydrodynamic context of viscous fingering. Our solutions
of the model [54, 78] show that the transition from convex to concave streamer head
indeed is dynamically possible (see Fig. 9); this is the onset of streamer branching. While
these solutions demonstrate the onset of branching, they have the unphysical property
that the local curvature of the boundary can become infinite within finite time. This
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Figure 9. Temporal evolution of the streamer in the moving boundary approximation
for an ideally conducting body, as shown in Fig. 8. The solution was computed
assuming that the electric potential accross the ionization boundary is continuous
= + and using conformal mapping methods [78].
cusp formation is suppressed in viscous fingering by a regularizing boundary condition.
Our analysis [79] of streamer fronts suggests a new boundary condition
+ = F(|E+|), F(|E+|) |E+| for |E+| > 2, (11)
for the potential jump accross the boundary. This boundary condition aproximates the
pdes (3)(5); it can be understood as a floating potential on the non-ionized side of the
ionization boundary, if the potential on the ionized side is fixed. First results [79] with
purely analytical methods indicate that this boundary condition indeed prevents cusp
formation, i.e., it regularizes the problem.
We conclude that streamer branching is generic even for deterministic streamermodels when they approach a state when the width of the space charge layer is much
smaller than its radius of curvature, as in the second and third column of Fig. 5. A sketch
of the distribution of surface charges and field is given by Fig. 8. A streamer in this
state is likely to branch due to a Laplacian instability. A more precise characterization
of the unstable state of the streamer head is under way.
5.3. How branching works and how it doesnt
It came as a surprise to many that a fully deterministic model like our fluid model would
exhibit branching, since another branching concept based on old pictures of Raether [12]is very well known. It is illustrated in Fig. 10. We remark on this concept:
1) The distribution and shape of avalanches ahead of the streamer head as shown in
Fig. 10 c to the best of our knowledge have never been substantiated by further analysis.
2) Even if this avalanche distribution is realized, it has not been shown that it would
evolve into several new streamer branches. Our major point of critique is that a space
charge distributed over the full streamer head as in the figure would be self-stabilizing
and not destabilizing, cf. a comparable recent analysis [81].
On the contrary, our main statement is: The formation of a thin space charge layer
is necessary for streamer branching while stochastic fluctuations are not necessary.
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The multiscale nature of streamers 17
(a)
(b)
(c)
Figure 10. (a) Space charge distribution in the emerging streamer head and (b)
streamer propagation by photoionization according to Raether 1939 [ 12]. Fig. (b)
was reproduced identically in [13] with English labels. Figure (c) shows the pictorial
concept of streamer branching by rare and long-range photoionization events and
successive avalanches as it can be found in many textbooks like [4]. It is a minor
variation of Fig. b with the avalanches not aligned, but placed around the streamer
head. The present version of the figure is taken from [11] where it motivates the
dielectric breakdown model [10]. Find our critique of this concept in the text.
Furthermore, for the question whether branching is possible in a deterministic fluid
model, we remind the reader that chaos is possible in fully deterministic nonlinear
models when their evolution approaches bifurcation points. Moreover, tip splitting in
Laplacian growth problems as described in Fig. 8 is well established in viscous fingering
in two fluid flow and other branches of physics.
5.4. The importance of charge transport and a remark on dielectric breakdown modelsWe have identified a state of the streamer head where it can branch. This state is
characterized by a weakly curved ionization front, i.e., the radius of curvature is much
larger than the width of the front. The width of the front is determined by the field E+
ahead of it [65], for high fields the width saturates [65, 77]. The formation of such a
weakly curved front requires a sufficiently high potential difference between streamer tip
and distant electrode and an appropriate charge content of the streamer head. Charge
is a conserved quantity. The consideration of charge conservation misses in the streamer
concepts suggested in [70, 71, 72].
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These considerations lead us to the general idea that a streamer tip is characterized
by electric potential , curvature R, field enhancement E+ and total charge content Q.
Only two of these four parameters are independent.
In contrast, dielectric barrier models (DBM) [10, 11] for multiply branched discharge
structures are characterized only by potential and longitudinal spatial structure. We
argue that a quantitative DBM model should also include the width of the streamer
channel and the related charge content as a model variable. This would allow an
appropriate characterization of streamer velocity and branching probability.
We remark that studies of streamer width and charge content are also crucial for
determining the electrostatic interaction of streamers or of leaders as their large
relatives.
6. Summary and outlook
The purpose of the present paper was to review the presently available methods to
investigate streamer discharges, in particular, those methods presently developed at TU
Eindhoven and CWI Amsterdam. The paper summarizes a talk given at the XXVIIth
International Conference on Phenomena in Ionized Gases (ICPIG) 2005.
Obviously, the complexity and the many scales of the phenomenon pose challenges
to experiments, simulations, modeling and analytical theory if one wants to proceed to
the quantitative understanding of more than a single non-branching streamer. In the
present stage, we have developped reliable methods in each discipline, they are reviewed
in the present paper. In the next stage, results of different methods should be compared:
simulations should be compared with experiments, and simulations should be checkedon consistency with analytical results. Analysis can also be used to extrapolate tediously
generated numerical results, once the emergence of larger scale coherent structures
like complete streamer heads with their inner layers has been demonstrated.
We have reviewed nanosecond resolved measurements of streamers and the
surprising influence of the power supply, streamer applications and the relation to
sprite discharges above thunderclouds. We have summarized the physical mechanisms of
streamer formation and a minimal continuum density model that contains the essentials
of the process. We have shown that a propagating and branching streamer even within
the minimal continuum model consists of very different length scales that can be
appropriately simulated with a newly developed numerical code with adaptive grids.
Finally, we have summarized our present understanding of streamer branching as a
Laplacian instability and compared it to earlier branching concepts.
On the experimental side, future tasks are precise measurements of streamer widths,
velocities and branching characteristics and their dependence on gas type and power sup-
ply as well as quantitative comparison of streamers and sprites. On the theoretical side,
both microscopic and macroscopic models should be developed further. Microscopically,
the particle dynamics in the limited region of the ionization front will be investigated in
more detail. Macroscopically, the quantitative understanding of streamer head dynam-
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The multiscale nature of streamers 19
ics should be incorporated into new dielectric breakdown models with predictive power.
In particular, charge transport and conservation should be included. The theoretical
tasks can only be treated succesfully, if a hierarchy of models on different length scales
is developed: from the particle dynamics in the streamer ionization front up to the
dynamics of a streamer head as whole. Elements of such a hierarchy are presented in
the present paper.
Acknowledgements: This paper summarizes work by a number of researchers
in a number of disciplines, therefore it has a number of authors. Beyond that, we
acknowledge inspiration and thought exchange with the pattern formation group of Wim
van Saarloos at Leiden Univ., with numerical mathematicians at cluster MAS at CWI
Amsterdam, with colleagues in physics and electroengineering at TU Eindhoven, as well
as with the many international colleagues and friends whom we met at conferences on
gas discharges, atmospheric discharges and nonlinear dynamics in physics and appliedmathematics. We thank Hans Stenbaek-Nielsen and Elisabeth Gerken for making sprite
figures 3 and 4 available.
The experimental Ph.D. work of Tanja Briels in Eindhoven is supported by a Dutch
technology grant (NWO-STW), the computational Ph.D. work of Carolynne Montijn is
supported by the Computational Science program of FOM and GBE within NWO. The
analytical Ph.D. work of Bernard Meulenbroek was supported by CWI. The postdoc
position of Andrea Rocco was payed by FOM-projectruimte and the Dutch research
school Center for Plasma Physics and Radiation Technology (CPS).
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